Subgroups — #33077317

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22621597	C₂	1	2	yes	maximal, complemented
3	#22609520	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609394	C₂	1	2	yes	maximal, complemented
5	#22609325	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609288	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609124	C₂	1	2	yes	maximal, complemented
8	#22609033	C₂	1	2	yes	maximal, complemented
9	#22608816	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924018	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#3391	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556621	—	2	2	no	—
13	D₄×Dic₁₄₆	C₂²	1	2	yes	—
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988991	—	2	1	no	—
16	#17988959	C₂²	1	2	yes	—
17	#17988933	C₂²	1	2	yes	—
18	#17988867	C₂²	1	2	yes	—
19	#17988865	—	2	2	no	—
20	#17988831	—	2	2	no	—
21	#17988797	C₂²	1	2	yes	—
22	#17988784	C₂²	1	2	yes	—
23	#17988749	—	2	2	no	—
24	#17988725	—	2	2	no	—
25	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
26	C₄×#11924018	C₂²	1	1	yes	characteristic
27	D₈×Dic₇₃	—	2	1	no	—
28	#17988665	C₂²	1	2	yes	—
29	#17988340	C₂²	1	1	yes	characteristic
30	#17988339	C₂²	1	1	yes	characteristic
31	#17988332	C₂²	1	2	yes	—
32	#17988331	C₂²	1	2	yes	—
33	Q₈⋊#10596831	C₂²	1	1	yes	characteristic
34	#17988317	C₂²	1	1	yes	characteristic
35	D₄⋊#10596831	C₂²	1	2	yes	—
36	C₈×#10596831	C₂²	1	1	yes	characteristic
37	#550×C₇₃	C₂²	1	2	yes	—
38	#493×C₇₃	—	2	1	no	—
39	#493×C₇₃	C₂²	1	1	yes	characteristic
40	#459×C₇₃	—	2	2	no	—
41	#459×C₇₃	C₂²	1	2	yes	—
42	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₇₃	—	2	1	no	—
44	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
45	C₂²×Dic₁₄₆	—	2	2	no	—
46	#14556621	—	4	8	no	—
47	C₂×#11924018	—	2	2	no	—
48	C₂×#11924018	D₄	1	1	yes	characteristic
49	C₂×#11924016	—	2	2	no	—
50	C₂×#11924016	—	2	2	no	—
51	C₂×#11924016	—	2	2	no	—
52	C₂×#11924012	—	2	2	no	—
53	#14556591	C₂³	1	2	yes	—
54	#14556590	—	2	2	no	—
55	#14556570	—	2	2	no	—
56	#14556568	—	4	2	no	—
57	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
58	D₄×Dic₇₃	—	2	2	no	—
59	#14556525	—	2	1	no	—
60	#14556525	—	2	1	no	—
61	#14556524	—	2	2	no	—
62	#14556524	D₄	1	1	yes	characteristic
63	C₄⋊#10596831	C₂³	1	1	yes	characteristic
64	C₄⋊#10596831	C₂³	1	2	yes	—
65	#14556520	—	2	1	no	—
66	#14556515	D₄	1	1	yes	characteristic
67	#14556514	—	2	2	no	—
68	#14556514	D₄	1	1	yes	characteristic
69	#14556512	—	2	2	no	—
70	#14556512	—	2	2	no	—
71	#14556512	—	4	2	no	—
72	C₂²⋊#10596831	—	2	2	no	—
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₈×Dic₇₃	—	2	1	no	—
76	C₂×#41×C₇₃	—	2	2	no	—
77	C₂×D₈×C₇₃	—	2	2	no	—
78	C₂×D₈×C₇₃	—	2	1	no	—
79	C₂×D₈×C₇₃	—	2	1	no	—
80	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
81	#127×C₇₃	—	2	2	no	—
82	#127×C₇₃	C₂³	1	2	yes	—
83	#121×C₇₃	—	2	2	no	—
84	#121×C₇₃	—	4	2	no	—
85	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₇₃	—	2	2	no	—
87	C₄×D₄×C₇₃	C₂³	1	2	yes	—
88	#107×C₇₃	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₇₃	—	2	1	no	—
90	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
91	#102×C₇₃	—	2	2	no	—
92	#102×C₇₃	D₄	1	2	yes	—
93	C₄×C₈×C₇₃	—	2	1	no	—
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₄₆	—	2	2	no	—
96	C₂×Dic₁₄₆	—	4	4	no	—
97	C₂×Dic₁₄₆	—	4	2	no	—
98	C₂×Dic₁₄₆	—	8	2	no	—
99	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
100	C₂²×Dic₇₃	—	4	2	no	—
101	#11924018	—	4	4	no	—
102	#11924018	D₈	1	4	yes	complemented
103	#11924016	—	4	4	no	—
104	#11924016	—	4	4	no	—
105	#11924016	—	8	2	no	—
106	#11924012	—	4	4	no	—
107	C₂×#10596831	—	2	1	no	—
108	C₂×#10596831	—	2	2	no	—
109	C₂×#10596831	—	2	1	no	—
110	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
111	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
112	C₄⋊Dic₇₃	—	2	2	no	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
115	#11924007	—	2	1	no	—
116	#11924007	—	4	2	no	—
117	C₂²⋊Dic₇₃	—	4	2	no	—
118	C₄×Dic₇₃	—	2	1	no	—
119	#41×C₇₃	—	4	4	no	—
120	#41×C₇₃	—	8	2	no	—
121	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₇₃	—	2	2	no	—
123	C₂×D₄×C₇₃	—	2	2	no	—
124	C₂×D₄×C₇₃	—	2	2	no	—
125	C₂×D₄×C₇₃	—	4	2	no	—
126	C₂×D₄×C₇₃	—	4	2	no	—
127	C₂×D₄×C₇₃	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₇₃	—	2	2	no	—
129	C₂²×C₄×C₇₃	—	4	2	no	—
130	D₈×C₇₃	—	2	2	no	—
131	D₈×C₇₃	—	4	2	no	—
132	D₈×C₇₃	—	8	1	no	—
133	D₈×C₇₃	—	8	2	no	—
134	C₂×C₈×C₇₃	—	2	1	no	—
135	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
137	C₄⋊C₄×C₇₃	—	2	1	no	—
138	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄×C₇₃	—	2	2	no	—
141	C₂²⋊C₄×C₇₃	—	4	2	no	—
142	C₄²×C₇₃	—	2	1	no	—
143	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
144	C₂³×C₇₃	—	2	2	no	—
145	C₂³×C₇₃	—	4	2	no	—
146	Q₈×C₇₃	—	2	1	no	—
147	Q₈×C₇₃	C₂×D₈	1	2	yes	—
148	D₄×C₇₃	—	2	2	no	—
149	D₄×C₇₃	—	2	2	no	—
150	D₄×C₇₃	—	4	2	no	—
151	D₄×C₇₃	—	4	2	no	—
152	D₄×C₇₃	—	4	2	no	—
153	D₄×C₇₃	—	4	2	no	—
154	D₄×C₇₃	—	8	4	no	—
155	D₄×C₇₃	—	8	2	no	—
156	D₄×C₇₃	—	16	2	no	—
157	C₂×C₄×C₇₃	—	2	1	no	—
158	C₂×C₄×C₇₃	—	2	1	no	—
159	C₂×C₄×C₇₃	—	2	2	no	—
160	C₂×C₄×C₇₃	—	4	2	no	—
161	C₂×C₄×C₇₃	—	4	2	no	—
162	C₂×C₄×C₇₃	—	8	4	no	—
163	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×Dic₇₃	—	2	1	no	—
167	C₂×Dic₇₃	—	2	1	no	—
168	C₂×Dic₇₃	—	4	2	no	—
169	C₂×Dic₇₃	—	8	4	no	—
170	Dic₁₄₆	—	2	1	no	—
171	Dic₁₄₆	—	4	2	no	—
172	Dic₁₄₆	—	4	2	no	—
173	Dic₁₄₆	—	8	4	no	—
174	Dic₁₄₆	C₂×D₈	1	2	yes	—
175	C₈×C₇₃	—	2	1	no	—
176	C₈×C₇₃	—	4	1	no	—
177	C₈×C₇₃	#137	1	2	yes	complemented
178	#10596831	—	2	1	no	—
179	#10596831	—	4	1	no	—
180	#10596831	—	4	2	no	—
181	#10596831	—	4	1	no	—
182	#10596831	C₂×D₈	1	2	yes	—
183	C₂²×C₇₃	—	4	2	no	—
184	C₂²×C₇₃	—	4	2	no	—
185	C₂²×C₇₃	—	4	2	no	—
186	C₂²×C₇₃	—	8	2	no	—
187	C₂²×C₇₃	—	8	2	no	—
188	C₂²×C₇₃	—	8	2	no	—
189	C₂²×C₇₃	D₄²	1	1	yes	characteristic
190	C₄×C₇₃	—	2	2	no	—
191	C₄×C₇₃	—	2	1	no	—
192	C₄×C₇₃	—	4	2	no	—
193	C₄×C₇₃	—	4	1	no	—
194	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
195	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
196	C₄×C₇₃	#576	1	1	yes	characteristic
197	C₄×C₇₃	#577	1	1	yes	characteristic
198	Dic₇₃	—	2	2	no	—
199	Dic₇₃	—	4	1	no	—
200	Dic₇₃	—	8	2	no	—
201	#19889	—	73	1	no	maximal, Sylow
202	C₂×C₇₃	—	8	2	no	—
203	C₂×C₇₃	—	16	2	no	—
204	C₂×C₇₃	#3278	1	1	yes	characteristic
205	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
206	C₂×C₇₃	#3290	1	1	yes	characteristic
207	#3391	D₇₃	1	1	yes	characteristic
208	Q₈×D₈	—	73	1	no	—
209	#3278	—	73	2	no	—
210	#1706	—	73	1	no	—
211	#1685	—	73	2	no	—
212	#1674	—	73	1	no	—
213	#1668	—	73	1	no	—
214	#1641	—	73	2	no	—
215	#1627	—	73	2	no	—
216	#1589	—	73	1	no	—
217	C₈×Q₁₆	—	73	1	no	—
218	C₇₃	#19889	1	1	yes	characteristic, complemented, Sylow
219	C₂×#137	—	146	2	no	—
220	#550	C₂×D₇₃	1	2	yes	—
221	D₄×Q₈	—	73	2	no	—
222	#500	—	73	1	no	—
223	#500	—	146	1	no	—
224	#493	—	2	1	no	—
225	#493	C₂×D₇₃	1	1	yes	characteristic
226	#492	—	73	2	no	—
227	#486	—	73	2	no	—
228	#474	—	73	2	no	—
229	#474	—	146	2	no	—
230	#468	—	146	2	no	—
231	#462	—	73	2	no	—
232	#460	—	73	2	no	—
233	#459	—	2	2	no	—
234	#459	C₂×D₇₃	1	2	yes	—
235	#452	—	146	2	no	—
236	#448	—	146	2	no	—
237	C₈×Q₈	—	73	1	no	—
238	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
239	C₄×Q₁₆	—	73	1	no	—
240	C₄×D₈	—	2	1	no	—
241	C₄×D₈	—	146	1	no	—
242	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
243	#436	—	73	2	no	—
244	#335	—	73	1	no	—
245	#335	—	73	1	no	—
246	#332	—	73	2	no	—
247	#331	—	73	2	no	—
248	Q₈⋊C₈	—	73	1	no	—
249	Q₈⋊C₈	—	73	1	no	—
250	D₄⋊C₈	—	73	2	no	—
251	C₈²	—	73	1	no	—
252	C₂×#41	—	2	2	no	—
253	C₂²×Q₈	—	146	2	no	—
254	#137	—	292	8	no	—
255	C₂×Q₁₆	—	73	1	no	—
256	C₂×Q₁₆	—	146	2	no	—
257	C₂×SD₁₆	—	146	2	no	—
258	C₂×SD₁₆	—	146	2	no	—
259	C₂×SD₁₆	—	146	2	no	—
260	C₂×D₈	—	2	2	no	—
261	C₂×D₈	—	2	1	no	—
262	C₂×D₈	—	2	1	no	—
263	C₂×D₈	C₂²⋊D₇₃	1	1	yes	characteristic
264	C₂×M₁₆	—	146	2	no	—
265	C₄⋊Q₈	—	73	2	no	—
266	C₄⋊Q₈	—	146	2	no	—
267	#127	—	2	2	no	—
268	#127	C₂²×D₇₃	1	2	yes	—
269	C₂²⋊Q₈	—	292	2	no	—
270	C₂²⋊Q₈	—	146	2	no	—
271	#121	—	2	2	no	—
272	#121	—	4	2	no	—
273	C₄×Q₈	—	73	1	no	—
274	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
275	C₄×D₄	—	2	2	no	—
276	C₄×D₄	—	146	2	no	—
277	C₄×D₄	C₂²×D₇₃	1	2	yes	—
278	#107	—	73	1	no	—
279	#107	—	146	1	no	—
280	#107	—	146	1	no	—
281	#107	—	146	2	no	—
282	#107	C₂²⋊D₇₃	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	73	2	no	—
285	C₄⋊C₈	—	73	1	no	—
286	C₄⋊C₈	—	146	1	no	—
287	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
288	#103	—	73	1	no	—
289	#103	—	73	1	no	—
290	#103	—	146	2	no	—
291	#102	—	2	2	no	—
292	#102	—	292	2	no	—
293	#102	—	146	2	no	—
294	#102	—	146	2	no	—
295	#102	C₂²⋊D₇₃	1	2	yes	—
296	C₂²⋊C₈	—	146	2	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	—	73	1	no	—
299	C₄×C₈	—	73	1	no	—
300	C₄×C₈	—	146	1	no	—
301	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
302	#41	—	4	4	no	—
303	#41	—	8	2	no	—
304	C₂×Q₈	—	292	2	no	—
305	C₂×Q₈	—	292	4	no	—
306	C₂×Q₈	—	584	2	no	—
307	C₂×Q₈	—	73	1	no	—
308	C₂×Q₈	—	146	2	no	—
309	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	C₂×C₂²⋊D₇₃	1	2	yes	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	292	2	no	—
319	Q₁₆	—	292	4	no	—
320	Q₁₆	—	73	4	no	—
321	SD₁₆	—	292	4	no	—
322	SD₁₆	—	292	4	no	—
323	SD₁₆	—	584	2	no	—
324	D₈	—	2	2	no	—
325	D₈	—	4	2	no	—
326	D₈	—	8	1	no	—
327	D₈	—	8	2	no	—
328	M₁₆	—	292	4	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	73	1	no	—
331	C₂×C₈	—	73	1	no	—
332	C₂×C₈	—	146	2	no	—
333	C₂×C₈	—	146	1	no	—
334	C₂×C₈	—	146	1	no	—
335	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
336	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	292	2	no	—
339	C₄⋊C₄	—	73	1	no	—
340	C₄⋊C₄	—	146	2	no	—
341	C₄⋊C₄	—	146	2	no	—
342	C₄⋊C₄	—	146	1	no	—
343	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
344	C₄⋊C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	292	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	146	1	no	—
350	C₄²	C₂³×D₇₃	1	1	yes	characteristic
351	C₂³	—	2	2	no	—
352	C₂³	—	4	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	292	2	no	—
355	Q₈	—	292	2	no	—
356	Q₈	—	584	4	no	—
357	Q₈	—	73	2	no	—
358	Q₈	—	146	1	no	—
359	Q₈	D₈×D₇₃	1	2	yes	—
360	D₄	—	2	2	no	—
361	D₄	—	2	2	no	—
362	D₄	—	4	2	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	2	no	—
366	D₄	—	8	4	no	—
367	D₄	—	8	2	no	—
368	D₄	—	16	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	292	2	no	—
376	C₂×C₄	—	584	4	no	—
377	C₂×C₄	—	146	1	no	—
378	C₂×C₄	—	146	1	no	—
379	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
381	C₂×C₄	#14556636	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	4	1	no	—
384	C₈	—	292	1	no	—
385	C₈	—	292	1	no	—
386	C₈	—	292	2	no	—
387	C₈	—	73	2	no	—
388	C₈	—	146	1	no	—
389	C₈	#14556621	1	2	yes	complemented
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	D₄×C₂²⋊D₇₃	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	2	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	292	1	no	—
402	C₄	—	584	2	no	—
403	C₄	—	146	2	no	—
404	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
405	C₄	C₂×#14556621	1	1	yes	characteristic
406	C₄	#17989388	1	1	yes	characteristic
407	C₄	#17989398	1	1	yes	characteristic
408	C₂	—	8	2	no	—
409	C₂	—	16	2	no	—
410	C₂	#22621597	1	1	yes	characteristic
411	C₂	D₈×C₂²⋊D₇₃	1	1	yes	characteristic
412	C₂	#22621713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting, prime radical	C₇₃×#3391	C₂
socle	C₂²×C₇₃	D₄²
hypercenter	#3391	D₇₃
hypocenter	C₇₃	#19889
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. D₈×Dic₁₄₆	C₂	semidirect
2. #22621597	C₂	semidirect
3. #22609520	C₂	semidirect
4. #22609394	C₂	semidirect
5. #22609325	C₂	semidirect
6. #22609288	C₂	semidirect
7. #22609124	C₂	semidirect
8. #22609033	C₂	semidirect
9. #22608816	C₂	semidirect
10. C₈×#11924018	C₂	semidirect
102. #11924018	D₈	semidirect
177. C₈×C₇₃	#137	semidirect
218. C₇₃	#19889	semidirect
389. C₈	#14556621	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₈×Dic₁₄₆	2
#22621597	2
#22609520	2
#22609394	2
#22609325	2
#22609288	2
#22609124	2
#22609033	2
#22608816	2
C₈×#11924018	2
C₇₃×#3391	2
#19889	73